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Periods of periodic points of 1-norm nonexpansive maps

Lemmens, Bas (2003) Periods of periodic points of 1-norm nonexpansive maps. Mathematical Proceedings of the Cambridge Philosophical Society, 135 (1). pp. 165-180. ISSN 0305-0041. (doi:10.1017/S0305004103006741) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:24186)

The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided. (Contact us about this Publication)
Official URL
http://dx.doi.org/10.1017/S0305004103006741

Abstract

In this paper several results concerning the periodic points of 1-norm non-expansive maps will be presented. In particular, we will examine the set $R(n)$, which consists of integers $p\geq 1$ such that there exist a 1-norm nonexpansive map $f{:}\ \mathbb{R}^n\rightarrow\mathbb{R}^n$ and a periodic point of $f$ of minimal period $p$. The principal problem is to find a characterization of the set $R(n)$ in terms of arithmetical and combinatorial constraints. This problem was posed in [12, section 4]. We shall present here a significant step towards such a characterization. In fact, we shall introduce for each $n\in\mathbb{N}$ a set $T(n)$ that is determined by arithmetical and combinatorial constraints only, and prove that $R(n)\subset T(n)$ for all $n\in \mathbb{N}$. Moreover, we will see that $R(n)=T(n)$ for $n=1,2,3,4,6,7$, and 10, but it remains an open problem whether the sets $R(n)$ and $T(n)$ are equal for all $n\in \mathbb{N}$.

Item Type: Article
DOI/Identification number: 10.1017/S0305004103006741
Subjects: Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Bas Lemmens
Date Deposited: 29 Jan 2013 14:27 UTC
Last Modified: 06 Feb 2020 04:05 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/24186 (The current URI for this page, for reference purposes)
Lemmens, Bas: https://orcid.org/0000-0001-6713-7683
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